Graffiti Conjecture 806

Let $S$ be the set of square-free integers in $[2, n]$ and $G = PR[S]$ the graph on $S$ in which two integers are adjacent when they are not coprime. From the cases $n \le 100$ and about twenty further values $n \le 200$, Graffiti conjectured that the largest adjacency eigenvalue $\lambda_1(G)$ is at most the number of distinct vertex degrees. False. At $n = 51$ the graph has $31$ vertices, $11$ distinct degrees and $\lambda_1 > 11.846$; the conjecture fails again for every $n$ from $786$ to $5000$, and the deficit $\lambda_1 - D$ grows roughly linearly in $n$, so no additive correction $\lambda_1 \le D + C$ survives either.

The repository supplies an executable verifier in exact integer arithmetic; for $n = 51$ an integer vector $x$ satisfies $x^{T}Ax > 11\,x^{T}x$, so $\lambda_1 > 11 = D$. The deficit $\lambda_1(n) - D(n)$ grows through $n = 5000$ in the repository's computations, but no asymptotic theorem proving divergence is claimed, so "false for every constant $C$" is a computed pattern, not a proved one. One anomaly, which the repository records itself: $n = 51$ lies inside the range Graffiti is said to have tested when it made the conjecture.